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FOCS
2009
IEEE

Linear Systems over Composite Moduli

13 years 10 months ago
Linear Systems over Composite Moduli
We study solution sets to systems of generalized linear equations of the form ℓi(x1, x2, · · · , xn) ∈ Ai (mod m) where ℓ1, . . . , ℓt are linear forms in n Boolean variables, each Ai is an arbitrary subset of Zm, and m is a composite integer that is a product of two distinct primes, like 6. Our main technical result is that such solution sets have exponentially small correlation, i.e. exp − Ω(n) , with the boolean function MODq, when m and q are relatively prime. This bound is independent of the number t of equations. This yields progress on limiting the power of constant-depth circuits with modular gates. We derive the first exponential lower bound on the size of depth-three circuits of type MAJ ◦ AND ◦ MODA m (i.e. having a MAJORITY gate at the top, AND/OR gates at the middle layer and generalized MODm gates at the base) computing the function MODq. This settles a decade-old open problem of Beigel and Maciel [5], for the case of such modulus m. Our technique mak...
Arkadev Chattopadhyay, Avi Wigderson
Added 20 May 2010
Updated 20 May 2010
Type Conference
Year 2009
Where FOCS
Authors Arkadev Chattopadhyay, Avi Wigderson
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